Editing Time preference (section)

== Models of time preference—time discounting ==
'''Temporal discounting''' (also known as '''delay discounting''', '''time discounting''')<ref name="Doyle2013" /> is the tendency of people to discount rewards as they approach a temporal horizon in the future or the past (i.e., become so distant in time that they cease to be valuable or to have addictive effects). To put it another way, it is a tendency to give greater value to rewards as they move away from their temporal horizons and towards the "now". For instance, a [[nicotine]] deprived smoker may highly value a cigarette available any time in the next 6 hours but assign little or no value to a cigarette available in 6 months.<ref name="Bickel1999" />

Regarding terminology, from Frederick et al. (2002):
{{blockquote|We distinguish ''time discounting'' from ''time preference''. We use the term ''time discounting'' broadly to encompass any reason for caring less about a future consequence, including factors that diminish the expected utility generated by a future consequence, such as uncertainty or changing tastes. We use the term ''time preference'' to refer, more specifically, to the preference for immediate utility over delayed utility.}}

This term is used in intertemporal economics, [[intertemporal choice]], [[reward system|neurobiology of reward]] and [[Decision making#Neuroscience perspective|decision making]], [[microeconomics]] and recently [[neuroeconomics]].<ref name=Takahashi2009 /> Traditional models of economics assumed that the discounting function is [[exponential discounting|exponential]] in time leading to a monotonic decrease in preference with increased time delay; however, more recent neuroeconomic models suggest a [[hyperbolic discounting|hyperbolic discount function]] which can address the phenomenon of preference reversal.<ref name="Green2004" />
Temporal discounting is also a theory particularly relevant to the political decisions of individuals, as people often put their short term political interests before the longer term policies.<ref>Schafer2016</ref> This can be applied to the way individuals vote in elections but can also apply to how they contribute to societal issues like climate change, that is primarily a long term threat and therefore not prioritized.<ref>{{cite journal | url=https://link.springer.com/content/pdf/10.1007/s11027-010-9239-9.pdf | doi=10.1007/s11027-010-9239-9 | title=Space-time discounting in climate change adaptation | date=2010 | last1=Baum | first1=Seth D. | last2=Easterling | first2=William E. | journal=Mitigation and Adaptation Strategies for Global Change | volume=15 | issue=6 | pages=591–609 }}</ref>

There have been many mathematical models of time preference that attempt to explain intertemporal preferences.

=== Exponential discounted utility ===
Exponential discounted utility was first described in the discounted utility model. The equation is as follows:

<math>U^t(c_t, \dots, c_T) = \sum_{k=0}^{T-t} D(k) u(c_{t+k})</math>

where

<math>D(k) = \left(\frac{1}{1 + \rho}\right)^k
</math>

is commonly thought of as the discount function, with <math>\rho</math>  being the discount rate.<ref name=":0" /> It says that your value of the future is exponentially less than your value of the present, as scaled by <math>\rho</math> and <math>k</math> . Although the equation was never meant to be ''normative'', ie, making a recommendation as to how people behave, it was the first template for modeling utility over time. Later, its ''descriptive'', validity, or ability to describe how people actually behave, was evaluated. Inconsistencies led to the theorizing of a new equation for time preference.

=== Hyperbolic discounting ===
Although the exponential equation provides a nice rationale for discounting in accordance with utility theory, the apparent rate, when measured in the lab, is not constant. It actually declines over time. This means that the difference between receiving $10 tomorrow and $11 in two days is different from receiving $10 in 100 days and $11 in 101 days. Although the difference between the values and the times is the same, people ''value'' the two options at a different discount rate. The $1 is more heavily discounted between tomorrow and two days than it is between 100 and 101 days, meaning that people prefer the $10 option more in the two day case than in the 100 day case.

Such preferences fit a hyperbolic curve. The first hyperbolic delay function was of the form<ref name=":0" />

<math>D(k) = 
\begin{cases} 
1 & \text{if } k = 0 \\ 
\beta \delta^k & \text{if } k > 0 
\end{cases}</math>

This function describes a difference between the discount rate today and the next period, and then constant discounting after. It is commonly called the <math>(\beta, \delta)</math> model.

A simple hyperbolic delay discounting equation is that of

<math>\frac{v}{V} = \frac{1}{1 + kD}</math>

Where <math>v</math> is the discounted value, <math>V</math> is the non-discounted value, <math>k</math> is the discount rate, and <math>D</math> is the delay.<ref>{{Cite journal |last=Rachlin |first=Howard |date=May 2006 |title=Notes on Discounting |journal=Journal of the Experimental Analysis of Behavior |language=en |volume=85 |issue=3 |pages=425–435 |doi=10.1901/jeab.2006.85-05 |issn=0022-5002 |pmc=1459845 |pmid=16776060}}</ref> This is one of the most common hyperbolic discounting functions used today, and is especially useful in comparing two discounting scenarios, as the <math>k</math> parameter can be easily interpreted.

=== Quasi-hyperbolic discounting ===
The last major model is that of quasi-hyperbolic discounting. Researchers found that there is a first day effect, meaning that people greatly value immediate rewards over those in the future. Like the previous example, imagine now that you are offered $10 today or $11 tomorrow. You are also offered $10 tomorrow or $11 in two days. The preference for the $10 in the today case is typically greater than the preference for the $10 tomorrow case.

This can be captured by a quasi-hyperbolic curve, wherein there is a fitted parameter for the magnitude of the first day effect. This is commonly called the beta-delta model, wherein there is a beta parameter that accounts for the present bias. The equation for utility over time looks like<ref>{{Cite journal |last1=O'Donoghue |first1=Ted |last2=Rabin |first2=Matthew |date=1999 |title=Doing It Now or Later |url=https://www.jstor.org/stable/116981 |journal=The American Economic Review |volume=89 |issue=1 |pages=103–124 |doi=10.1257/aer.89.1.103 |jstor=116981 |issn=0002-8282|url-access=subscription }}</ref>

<math>U_t(u_t, u_{t+1}, \dots, u_T) = \delta^t u_t + \beta \sum_{s=t+1}^T \delta^{s-t} u_s
</math>

This explains that the sum of your current and all future utilities is equal to a delta parameter multiplied by your current utility plus all your future discounted utilities (scaled by beta).